Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$.

How can I prove these two statements? I guess I should use the fact that every column of unitary matirx is orthonormal, but I'm not sure where to put that...

share|improve this question

1 Answer 1

up vote 2 down vote accepted

We have $\det U^H=\overline{\det U}$ and $\det(AB)=\det A\det B$ for two square matrices of the same dimension, which gives $$1=\det I_n=\det U\cdot U^H=\det U\cdot\det U^H=\det U\overline{\det U}=|\det U|^2.$$

We can have $\det U=\det U^H$, when $U=I_n$ for example.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.